This is an elaborate study of global qualitative results for dissipative systems with constraints. The authors consider a compact, connected, Riemannian configuration manifold M, a (perfect) constraint subbundle \(\Sigma\) M of TM and a force field F: TM\(\to T^*M\), consisting of a conservative and a dissipative part: \(F=dV+D\). They first establish a general theory concerning the reactions corresponding to perfect constraints and the construction of the Gibbs-Maggi-Appell (GMA) vector field on \(\Sigma\) M.
In section 3, some deep results are obtained in studying the attractor \({\mathcal A}\), consisting of all globally defined and bounded orbits on \(\Sigma\) M. It is shown, for example, that the \(\alpha\) and \(\omega\)- limit sets of any orbit in \({\mathcal A}\) are points, under the assumption that the center manifold of each equilibrium coincides locally with the set of equilibria. In that case, moreover, \({\mathcal A}\) appears to be the union of the unstable manifolds of the equilibrium points.
In section 4, the relationship between the attractor \({\mathcal A}\) and the configuration space M is studied. It is shown that both have the same dimension if V is a Morse function, D is strongly dissipative and \({\mathcal A}\) is a manifold. For sufficiently small V, moreover, \({\mathcal A}\) is diffeomorphic to M. The final section concerns the study of the flow of the GMA vector field restricted to \({\mathcal A}\), with respect to a corresponding flow on M.